Mock AIME II 2012 Problems/Problem 1

Problem

Given that $\left(\dfrac{6^2-1}{6^2+11}\right)\left(\dfrac{7^2-2}{7^2+12}\right)\left(\dfrac{8^2-3}{8^2+13}\right)\cdots\left(\dfrac{2012^2-2007}{2012^2+2017}\right)=\dfrac{m}{n},$ where $m$ and $n$ are positive relatively prime integers, find the remainder when $m+n$ is divided by $1000$ .

Solution

Consider $\frac{k^2-(k-5)}{k^2+(k+5)}$ . We note that $(k+1)^2-(k+1-5)=k^2+(k+5)$ , thus we have a telescoping sequence and we need only consider the first numerator and last denominator.

$\frac{6^2-1}{2012^2+2017}$

Note that $5|(6^2-1)$ however $2012^2+2017\equiv 4+2\equiv 1\pmod{5}$ . Also, note that $7|(6^2-1)$ however $2012^2+2017\equiv 25+4\equiv 1\pmod{7}$ . Since $6^2-1=5*7$ , we know that $\gcd(6^2-1, 2012^2+2017)=1$ . Now note that we want $35+2012^2+2017 \pmod{1000}$ , therefore we use $2012^2\equiv 12^2\pmod{1000}$ and $2017\equiv 17\pmod{1000}$ to give us $m+n=35+144+17=\boxed{196}\pmod{1000}$ .

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Mock AIME II 2012 Problems/Problem 1

Problem

Solution